Before playing professionally, Adcock attended North Hardin High School. He was drafted by the Seattle Mariners in the fifth round of the 2006 Major League Baseball Draft and began his professional career that year.

With the AZL Mariners in 2006, Adcock went 0-2 with a 3.31 ERA in 10 games (three starts). In 2007, he went 3-11 with a 4.58 ERA in 22 games (21 starts) for the Wisconsin Timber Rattlers and High Desert Mavericks. He pitched for Wisconsin again in 2008, going 2-5 with a 3.72 ERA in 15 games (14 starts), striking out 82 batters in 77 1/3 innings. He began the 2009 season with High Desert, however on July 29, 2009, he was traded with Ronny Cedeño, Jeff Clement, Aaron Pribanic and Brett Lorin to the Pittsburgh Pirates for Ian Snell and Jack Wilson.

On July 29, 2009, he was traded with Ronny Cedeño, Jeff Clement, Aaron Pribanic and Brett Lorin to the Pittsburgh Pirates for Ian Snell and Jack Wilson. He finished the year with the Lynchburg Hillcats and went 8-9 with a 5.29 ERA in 28 games (23 starts) that season. In 2010, he went 11-7 with a 3.38 ERA in 27 games (26 starts) for the Bradenton Marauders. He was taken by the Kansas City Royals in the 2010 Rule 5 Draft.

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